Kepler's first law
Gravitational Fields - OCR A-Level Physics
- Kepler's first lawAll planets orbit the Sun in ellipses with the Sun at one focus.: Planets move in elliptical orbits with the Sun at one focus.
- Kepler's second lawA line joining a planet and the Sun sweeps out equal areas in equal times.: A line joining a planet to the Sun sweeps out equal areas in equal times. (Planets move faster when closer to the Sun.)
- Kepler's third lawThe square of the orbital period is proportional to the cube of the semi-major axis: T² ∝ r³.: $T^{2} \propto r^{3}$, or equivalently $T^{2}/r^{3}$ is constant for all objects orbiting the same central mass.
$$T^2 = \frac{4\pi^2}{GM}r^3$$
- Derived by equating gravitational force to centripetal forceThe resultant force directed towards the centre of a circular path that causes an object to move in a circle. It is not a separate force but the net force providing circular motion.: $GMm/r^{2} = m(2\pi/T)^2 r$.
- Rearranging: $T^{2} = (4\pi^2/GM)r^{3}$.
- This can be used to find the mass of the central body if $T$ and $r$ are known: $M = 4\pi^2 r^{3}/(GT^2)$.
- For circular orbits, the orbital speed is $v = \sqrt{GM/r}$. Higher orbits have slower speeds.
Examiner Tips and Tricks
- To find the mass of a planet from satellite data: measure $T$ and $r$, then use $M = 4\pi^2 r^{3} / (GT^2)$.
- This is how we determine the masses of planets and stars.